A new model of dry friction oscillator colliding with a rigid obstacle

Abstract

We consider a system of two masses connected by linear springs and in contact with a belt moving at a constant velocity. One of the masses can collide with a fixed rigid obstacle. The contact forces between the masses and the belt are given by Coulomb’s laws. Moreover, when the colliding mass is in contact with the obstacle, we assume that a perfect elastic impact occurs. Several periodic orbits including contact against the fixed obstacle followed by slip and stick phases are obtained in analytical form.

1 Introduction

Non-smooth dynamical systems are related to force or motion characteristics which are non-continuous. One example of them is dry friction oscillators. Another kind of non-smooth systems is vibrating systems with clearance between the moving parts. In many industrial applications like brake systems, machine tools or turbo machines, the combined actions of dry friction and impact induce some undesirable effects. Non-smooth systems are very complex and they are usually modeled as spring mass oscillators. In the past, such systems have been the subject of several investigations, mainly in the case of one-degree-of-freedom systems [1,2,3,4,5,6,7,8]. For multidegrees-of-freedom systems, very often, only numerical methods have been used [9,10,11,12]. However, in [13], a two-degree-of-freedom oscillator with a colliding component is considered and several results about the existence of periodic motions are obtained in analytical form. On the other hand, double dry friction oscillators have been considered in [14, 15]. Assuming that the friction forces are modeled by Coulomb’s laws, closed-form solutions including stick–slip phases are presented. More recently, in [16], a two-degree-of-freedom vibro-impact system with multiple constraints has been investigated by using the flow switch ability theory of discontinuous systems, while in [17, 18], mathematical investigation of a dry friction oscillator in contact with a speed-varying traveling belt or submitted to a switching control law has been performed.

In this paper, a two-degree-of-freedom oscillator excited by dry friction and in the presence of a rigid obstacle is considered. The existence of periodic orbits including an impact with the fixed obstacle and several phases of stick and slip motions is proved.

2 Description of the model

The system (Fig. 1) consists of two masses ${m}_1 ,{m}_2$ connected by linear springs of stiffness ${k}_1 ,{k}_2$. The two masses are in contact with a driving belt moving at a constant velocity $\nu _0$. Friction forces ${F}_1 ,{F}_2$ act between the masses and the belt. Moreover, the second mass can collide with a fixed rigid obstacle.

Fig. 1

Description of the model

Two different cases will be considered:

2.1 Motion impact-less case

This case has been investigated in [14]. The motion equations are given by:

$$\begin{aligned}&{M}\ddot{{y}}+{Ky}={F},\,\,{y}=\left( {{y}_1,{y}_2 } \right) ^{{t}},\,\,{F}=\left( {{F}_1,{F}_2 } \right) ^{{t}} \nonumber \\&{M}=\left( {\begin{array}{ll} {m}_1 &{} \quad 0 \\ 0 &{} \quad {m}_2 \\ \end{array}} \right) , \quad {K}=\left( {\begin{array}{ll} {k}_1 +{k}_2 &{} \quad -{k}_2 \\ -{k}_2 &{} \quad -{k}_2 \\ \end{array}} \right) , \nonumber \\&\qquad {y}_2 <{e}, \quad \ddot{{y}}=\frac{{\hbox {d}}^{2}{y}}{{\hbox {d}t}'^{2}} \end{aligned}$$

(1)

Here (${y}_1 ,{y}_2$) are the displacements of the masses, e is the clearance ,and (${{F}_1 ,{F}_2}$) are the contact friction forces obtained from Coulomb’s laws:

• $\nu _0 -\dot{{y}}_i \ne 0,\quad {F}_{{i}} = {F}_{\mathrm{si}} \hbox {{sign}}(\nu _0 -\dot{{y}}_{{i}} )\, ({i}=1,2)$

• $\nu _0 -\dot{{y}}_1 =0$

  $$\begin{aligned} {F}_1 \,{=}\left\{ {\begin{array}{l} ({k}_1 {+}{k}_2 ){y}_1 -{k}_2 {y}_2 \,\, \hbox {{if}}\,\, \left| {({k}_1+{k}_2 ){y}_1 -{k}_2 {y}_2 } \right| {<}{F}_{{r}1}\\ \varepsilon {F}_{{s}1} \,\, \hbox {if} \,\, \varepsilon [({k}_1 +{k}_2 ){y}_1 -{k}_2 {y}_2 ]>{F}_{{r}1} (\varepsilon =\pm \,1) \\ \end{array}} \right. \end{aligned}$$

  (2)

• $\nu _0 -\dot{{y}}_2 =0$

  $$\begin{aligned} {F}_2 =\left\{ {\begin{array}{l} {k}_2 ({y}_2 -{y}_1 ) \quad \hbox {if} \,\, \left| {{k}_2 ({y}_2 -{y}_1 )} \right| <{F}_{{r}2}\\ \varepsilon {F}_{{s}2} \quad \hbox {if}\,\, \varepsilon {k}_2 ({y}_2 -{y}_1 )>{F}_{{r}2} (\varepsilon =\pm \,1)\\ \end{array}} \right. \\ \end{aligned}$$

$F_{s1}, F_{s2}$ are the friction forces when slip motion occurs, while $F_{r1} ,F_{r2}$ are the static friction forces ($F_{si} <F_{ri}$).

The systems (1), (2) are normalized using:

$$\begin{aligned} {t}= & {} \omega _3 {t}',\omega _3 =\sqrt{\frac{{k}_1 +{k}_2 }{{m}_1 }}, \quad ({o}')=\frac{{\hbox {d}}({o})}{{\hbox {d}t}},\nonumber \\ {x}_{{i}}= & {} \frac{{y}_{{i}} }{{e}},({i}=1,2),{V}=\frac{\nu _0}{{e}\omega _3 } \end{aligned}$$

(3)

From (1), it follows:

$$\begin{aligned} {x}''+\tilde{{K}}{x}= & {} {R},{x}=\left( {{x}_1 ,{x}_2 } \right) ^{{t}},{x}_2 <1 \nonumber \\ \tilde{{K}}= & {} \left( {\begin{array}{ll} 1 &{} \quad -\chi \\ -\chi \eta &{} \quad \chi \eta \\ \end{array}} \right) , \quad {R}=\left( {{u}_1 ,\eta {u}_2 } \right) ^{{t}}\nonumber \\ \chi= & {} \frac{{k}_2 }{{k}_1 +{k}_2 }, \quad \eta =\frac{{m}_1 }{{m}_2 },\nonumber \\ {u}_{{i}}= & {} \frac{{F}_{{i}} }{({k}_1 +{k}_2 ){e}}({i}=1,2) \end{aligned}$$

(4)

For each mass, three kinds of motions occur: slip motion with a velocity less than the belt velocity, overshoot motion with a velocity greater than the belt velocity and stick motion with a velocity equal to the belt one. For each kind of motion, the closed-form solution is available [14]. In the following, instead of the parameters (${x},{x'}$), a new set of variables is introduced:

$$\begin{aligned}&{Z}=\left( {\begin{array}{l} z \\ z'\\ \end{array}} \right) , \quad {z}={x}-{d}_0 , \quad {d}_0 =\left( {{d}_{01} ,{d}_{02} } \right) ^{{t}}\nonumber \\&{z}'={x}', \quad {d}_{01} =\frac{{u}_{{s}1} +{u}_{{s}2} }{1-\chi },\nonumber \\&{d}_{02}=\frac{\chi {u}_{{s}1} +{u}_{{s}2} }{\chi (1-\chi )}. \end{aligned}$$

(5)

2.2 Rigid impact

In the case of a contact of the second mass with the stop at ${t}=\tau$, i.e., ${x}_2 (\tau )=1$, the positions and the velocities ${x}^{-}(\tau ), x'^{-}(\tau )$ of the system before the impact and the positions and the velocities ${x}^{+}(\tau ), {x}'^{+}(\tau )$ after the impact are related by:

$$\begin{aligned} {x}^{+}(\tau )={x}^{-}(\tau ), \,\, {x}'^{+}(\tau )=\hbox {Ex}'^{-}(\tau ), \,\,{E}=\left( {\begin{array}{ll} 1 &{} \quad 0 \\ 0 &{} \quad -\,1 \\ \end{array}} \right) \nonumber \\ \end{aligned}$$

(6)

Several transitions between all these kinds of regimes (i.e., slip motion, overshooting motion, stick motion and contact) can occur. In [7, 8], a more simple system, with only one-degree-of-freedom has been investigated. Several periodic motions including a shock have been obtained. In the following, we show that a similar investigation can be performed for this more complex system.

Fig. 2

Phase portrait of the non-colliding mass

3 First example of periodic orbits with impact (symmetric solution)

Let us assume the following initial conditions:

$$\begin{aligned} {x}_{10} ,{x}_{20} =1,{x}'_{10} ,0<{x}'_{20} <{V} \end{aligned}$$

(7)

At ${t}=0$ an impact occurs. After the impact, the positions and the velocities of the system are obtained by the formula:

$$\begin{aligned}&{Z}_0^+ ={H}_0 {Z}_0 , \,\,{H}_0=\left( {\begin{array}{ll} {I} &{} \quad 0 \\ 0 &{} \quad {E} \\ \end{array}} \right) ,\nonumber \\&\quad {I}=\left( {\begin{array}{ll} 1 &{} \quad 0 \\ 0 &{} \quad 1 \\ \end{array}} \right) ,\,\,{Z}_0 ={Z}(0) \end{aligned}$$

(8)

For $0<{t}<{T}$, the system undergoes a slip–slip motion given by

$$\begin{aligned} {Z}({t})={H}({t}){Z}_0^+ , \,\,{H}({t})=\left( {\begin{array}{ll} {H}_1({t}) &{} \quad {H}_2 ({t}) \\ {H}_3({t}) &{} \quad {H}_1 ({t}) \\ \end{array}} \right) \end{aligned}$$

(9)

The two-by-two matrices ${H}_{{i}} ({t}),({i}=1, 2, 3)$ are obtained [14] from a modal analysis of system (4). A periodic solution of period T is obtained if

$$\begin{aligned} {Z}_0 ={H}({T}){Z}_0^+ ={H}({T}){H}_0 {Z}_0 \end{aligned}$$

(10)

From (10), we deduce:

$$\begin{aligned}&({H}_1 -{I}){z}_0 -{H}_2 {z}'_0 =0, \,\, -{H}_3 {z}_0 +({H}_1 -{E}){z}'_0 =0\nonumber \\&\quad {H}_{{i}} ={H}_{{i}} ({T}),({i}=1,2,3) \end{aligned}$$

(11)

Taking into account the following properties [13] of the ${H}_{{i}}$ matrices:

$$\begin{aligned} {H}_1^2 -{H}_2 {H}_3 ={I},\,{H}_{{i}} {H}_{{j}} ={H}_{{j}} {H}_{{i}} ,\,({i},{j}=1,2,3)\nonumber \\ \end{aligned}$$

(12)

(11) gives the relation:

$$\begin{aligned} {H}_2 ({E}+{I}){z}'_0 =0, \quad \hbox {if}\,\, \det ({H}_2 )\ne 0,({E}+{I}){z}'_0 =0 \end{aligned}$$

(13)

From (13), as in Ref. [13], we deduce the initial conditions for this orbit:

$$\begin{aligned} {z}'_{10} =0,{z}_0 =({H}_1 -{I})^{-1}{H}_2 {z}'_0 \end{aligned}$$

(14)

This periodic orbit depends on 3 parameters (${z}_{10}, {z}'_{20} ,{T}$) defined by 2 scalar equations deduced from (14). As in Ref. [13], the period T can be chosen, and the two parameters (${z}_{10} ,{z}'_{20}$) are obtained from (14) in term of the period. Moreover, as in [14], an interesting property of symmetry for the phase portraits of the system is obtained (see “Appendix”).

Example: For

$$\begin{aligned} \eta= & {} 4,\chi =.3,\,\,{T}=2,{u}_{{s}1} =0.5,\,\,{u}_{{s}2} =0.3,\nonumber \\ {V}= & {} 2,\,\,{u}_{{r}1} =0.6,\,\,{u}_{{r}2} =0.4 \end{aligned}$$

(15)

we obtain:

$$\begin{aligned} {z}_{10} =-\,0.5058,\,\,{z}_{20} =-\,1.1429,\,\,{z}'_{20} =1.3837 \end{aligned}$$

(16)

The phase portraits of (${m}_1 ,{m}_2$) are shown in Figs. 2 and 3.

Fig. 3

Phase portrait of the colliding mass

4 Second kind of periodic solutions with impact

Let us consider the following initial conditions:

$$\begin{aligned} {x}_{10}, {x}_{20}\,{<}\,1,\,{x}'_{10} \,{<}\,{V},\,{x}'_{20} ={V},\left| {\chi ({x}_{20} -{x}_{10} } \right| <{u}_{{r}2} \end{aligned}$$

(17)

For $0<{t}<\tau$, the system performs a slip–stick motion [14] given by

$$\begin{aligned} {Z}({t})=\varGamma ({t}){Z}_0 ,\varGamma ({t})=\left( {\begin{array}{ll} \varGamma _1 ({t}) &{} \varGamma _2 ({t}) \\ \varGamma _3 ({t}) &{} \varGamma _1 ({t}) \\ \end{array}} \right) \end{aligned}$$

(18)

This motion ends at ${t}=\tau$ if at this time

$$\begin{aligned} {x}_{2{B}} \equiv {x}_2 (\tau )=1 \end{aligned}$$

(19)

At ${t}=\tau$, an impact occurs and we have:

$$\begin{aligned} {Z}_{{B}}^+ ={H}_0 {Z}_{{B}} ,{Z}_{{B}} \equiv {Z}(\tau )=\varGamma (\tau ){Z}_0 \end{aligned}$$

(20)

For $0<{t}-\tau <\tau _1$, the system undergoes a slip–slip motion. A periodic motion of period

$$\begin{aligned} {T}= & {} \tau +\tau _1 \quad \hbox {is obtained if}\ {Z}_0 ={H}(\tau _1 ){Z}_{{B}}^+\nonumber \\= & {} {H}(\tau _1 ){H}_0 \varGamma (\tau ){Z}_0 \end{aligned}$$

(21)

This motion depends on 5 parameters (${z}_{10}, {z}_{20}, {z}'_{10} ,\tau ,\tau _1$). From (19), (21), we obtain 5 scalar equations for the determination of these parameters.

Example: For

$$\begin{aligned} \eta =4,\,\chi =0.3,\,{V}=1,\,{u}_{{r}1} =0.5, \nonumber \\ {u}_{{s}1} =0.1,\,{u}_{{r}2} =0.4,\,{u}_{{s}2} =0.0225 \end{aligned}$$

(22)

we obtain:

$$\begin{aligned} \tau= & {} 2,\tau _1 =2.5075,\,{z}_{10} =0.1015,{z}_{20}\nonumber \\= & {} -\,1.2502,\,{z}'_{10} =-\,0.2532 \end{aligned}$$

(23)

The phase portraits of ${m}_1$ and ${m}_2$ are shown in Figs. 4 and 5, and the curves

$$\begin{aligned}&{f}_1 =\chi ({z}_2{-}{z}_1 )\,{+}\,{u}_{{s}2} \,{-}\,{u}_{{r}2} ,\nonumber \\&{f}_2 =\chi ({z}_2\,{-}\,{z}_1 )\,{+}\,{u}_{{s}2} \,{+}\,{u}_{{r}2} \end{aligned}$$

(24)

connected to the constraints (${f}_1 \le 0,{f}_2 >0)$ during the sticking motion of ${m}_2$ are shown in Fig. 6.

Fig. 4

Phase portrait of the non-colliding mass

Fig. 5

Phase portrait of the colliding mass

Fig. 6

Constraints during the sticking motion of ${m}_2$

5 Third kind of periodic solutions with impact

Let us assume the following initial conditions:

$$\begin{aligned}&x'_{10} =x'_{20} =V,\,\,x_{10} -\chi x_{20} =u_{r1},\nonumber \\&\quad \chi (x_{20} -x_{10} )<-u_{r2} \end{aligned}$$

(25)

For $0<t<\tau$, the system performs a slip–overshoot motion:

($x'_1 <V,x'_2 >V,\,u_1 =u_{s1} ,\,u_2 =-u_{s2}$) given by [14]:

$$\begin{aligned} {Z}({t})= & {} {H}({t}){Z}_0 +2{u}_{{s}2} ({H}({t})-{I}_4 ){A}_0 \nonumber \\ {A}_0= & {} \left( {\begin{array}{l} \alpha _0 \\ 0 \\ \end{array}} \right) ,\alpha _0 =\left( {\begin{array}{l} 1/(1-\chi ) \\ 1/\chi (1-\chi ) \\ \end{array}} \right) ,{I}_4 =\left[ {\begin{array}{ll} {I} &{} \quad 0 \\ 0 &{} \quad {I} \\ \end{array}} \right] \nonumber \\ \end{aligned}$$

(26)

This motion ends at $t=\tau$, if

$$\begin{aligned} {x}'_{1{B}}< & {} {V},\,{x}'_{2{B}} ={V},\,-{u}_{{r}2}<\chi ({x}_{2{B}} -{x}_{1{B}} )<{u}_{{r}2} ,\nonumber \\ {x}_{{iB}}\equiv & {} {x}_{{i}} (\tau ),{x}'_{{iB}} \equiv {x}'_{{i}} (\tau ),({i}=1,2) \end{aligned}$$

(27)

At this time, we have:

$$\begin{aligned}&{Z}_{{B}} ={HZ}_0 +2{u}_{{s}2} ({H}-{I}_4 ){A}_0 ,{Z}_{{B}}\nonumber \\&\quad \equiv {Z}(\tau ),{H}\equiv {H}(\tau ) \end{aligned}$$

(28)

For $0<{t}-\tau <\tau _1$, the system performs a slip–stick motion (${x}'_1 <{V},{x}'_2 ={V},{u}_1 ={u}_{{s}1}$).

This motion ends at ${t}=\tau +\tau _1$ if at this time:

$$\begin{aligned} {x}_{2{C}}\equiv {x}_2 (\tau +\tau _1 )=1,\chi \left| {{x}_{2{C}} -{x}_{1{C}}} \right| <{u}_{{r}2} \end{aligned}$$

(29)

After the impact, the positions and the velocities of the system are obtained from:

$$\begin{aligned} {Z}_{{C}}^+ ={H}_0 {Z}_{{C}} ,{Z}_{{C}} =\varGamma {Z}_{{B}} ,\varGamma \equiv \varGamma (\tau _1) \end{aligned}$$

(30)

For $0<{t}-\tau -\tau _1 <\tau _2$, the system undergoes a slip–slip motion (${x}'_1<{V},{x}'_2 <{V},{u}_1 ={u}_{{s}1} ,{u}_2 ={u}_{{s}2}$)

This motion ends at ${t}=\tau +\tau _1 +\tau _2$ if at this time we have:

$$\begin{aligned} {x}'_{1{D}}\equiv & {} {x}'_1 (\tau +\tau _1 +\tau _2 )={V},\left| {{x}_{1{D}} -\chi {x}_{2{D}}} \right| <{u}_{{r}1} \nonumber \\ {Z}_{{D}}\equiv & {} {Z}(\tau +\tau _1 +\tau _2 )={hZ}_{{C}}^+ ,{h}\equiv {H}(\tau _2 ) \end{aligned}$$

(31)

For $0<{t}-\tau -\tau _1 -\tau _2 <\tau _3$, the system performs a stick–slip motion (${x}'_1 ={V},{x}'_2 <{V},{u}_2 ={u}_{{s}2}$) given by [14]:

$$\begin{aligned}&{Z}({t}-\tau -\tau _1 -\tau _2 )={C}({t}){Z}_{{D}},\nonumber \\&\quad {C}({t})=\left( {\begin{array}{ll} {C}_1 ({t}) &{} \quad {C}_2 ({t}) \\ {C}_3 ({t}) &{} \quad {C}_1 ({t}) \\ \end{array}} \right) \end{aligned}$$

(32)

A periodic solution of period ${T}=\tau +\tau _1+\tau _2 +\tau _3$ is obtained if

$$\begin{aligned} {Z}_{{D}}= & {} {C}(-\tau _3){Z}_0 ,{Z}_{{ D}} ={QZ}_0 +2{u}_{{s}2} ({Q}-{hH}_0) {A}_0\nonumber \\ {Q}= & {} {hH}_0 \varGamma {H} \end{aligned}$$

(33)

This solution depends on 6 parameters (${x}_{10} ,{x}_{20}, \tau ,\tau _1 ,\tau _2 ,\tau _3$). Taking into account that the condition (31) is included in the periodicity conditions (33), these parameters are linked by 7 scalar equations deduced from (25), (27), (29), (33).

We solve a semi-inverse problem, assuming that ${u}_{{r}1}$ is defined by (25). The solution is obtained by solving the 6 scalar equations deduced from (27), (29), (33) with respect to (${x}_{10} ,{x}_{20} ,\tau ,\tau _1 ,\tau _2 ,\tau _3$).

Example: For

$$\begin{aligned}&\eta =1,\,\chi =0.2,\,{V}=0.556,\,{u}_{{s}1} =0.0535, \nonumber \\&\quad {u}_{{r}1} =1.2018,\,{u}_{{s}2} =0.2161,\,{u}_{{r}2} =0.28 \end{aligned}$$

(34)

we obtain:

$$\begin{aligned}&\tau =1.5775,\,\tau _1 =1,\,\tau _2 =1.5,\,\tau _3 =3.281,\nonumber \\&{z}_{10}=0.7653,\,{z}_{20} =-\,1.9152 \end{aligned}$$

(35)

The phase portraits of ${m}_1$ and ${m}_2$ are shown in Figs. 7 and 8. Moreover, the constraints during the sticking motion of ${m}_2 (0<{t}-\tau <\tau _1$) and the constraints during the sticking motion of ${m}_1 (0<{t}-\tau -\tau _1 -\tau _2 <\tau _3$) are fulfilled.

Fig. 7

Phase portrait of the non-colliding mass

Fig. 8

Phase portrait of the colliding mass

Fig. 9

Phase portrait of the non-colliding mass

6 Fourth kind of periodic solutions with impact

Let us consider the following initial conditions:

$$\begin{aligned} {x}'_{10}= & {} {x}'_{20} ={V},{x}_{20}=1,{x}_{10} -\chi {x}_{20}\nonumber \\< & {} -{u}_{{r}1} ,\chi \left| {{x}_{20} -{x}_{10} } \right| <{u}_{{r}2} \end{aligned}$$

(36)

At ${t}=0$, an impact of the second mass occurs with the post-impact rule:

$$\begin{aligned} {Z}_0^+ ={H}_0 {Z}_0 \end{aligned}$$

(37)

For $0<{t}<\tau$, the system performs an overshoot–slip motion (${x}'_1>{V},{x}'_2<{V}$). This motion ends at ${t}=\tau$ if at this time, we have:

$$\begin{aligned} {x}'_{1{B}} \equiv {x}'_1 (\tau )={V} \end{aligned}$$

(38)

The positions and the velocities at ${t}=\tau$ are obtained from the formula [14]:

$$\begin{aligned} {Z}_{{B}}\equiv & {} {Z}(\tau )={H}(\tau ){Z}_0^+ +2{u}_{{s}1} ({H}(\tau )-{I}_4 ){B}_0 \nonumber \\&{B}_0 =\left( {\begin{array}{l} \beta _0 \\ 0 \\ \end{array}} \right) ,\beta _0 =\left( {\begin{array}{l} 1/(1-\chi ) \\ 1/(1-\chi ) \\ \end{array}} \right) \end{aligned}$$

(39)

Let us assume the following properties:

$$\begin{aligned} \left| {{x}_{1{B}} -\chi {x}_{2{B}} } \right|<{u}_{{r}1} ,{x}'_{2{B}} <{V} \end{aligned}$$

(40)

For $0<{t}-\tau <\tau _1$, the system undergoes a stick–slip motion (${x}'_1 ={V},{x}'_2<{V}$)

This motion ends at ${t}=\tau +\tau _1$ if at this time, we have:

$$\begin{aligned} {x}_{1{C}} -\chi {x}_{2{C}} ={u}_{{r}1}\ \hbox {{where}}\ {Z}_{{C}} \equiv {Z}(\tau {+}\tau _1 )={C}(\tau _1 ){Z}_{{B}}\nonumber \\ \end{aligned}$$

(41)

For $0<{t}-\tau -\tau _1 <\tau _2$, the system performs a slip–slip motion (${x}'_1<{V},{x}'_2<{V}$). This motion ends at ${t}=\tau +\tau _1 +\tau _2$ if at this time, we have:

$$\begin{aligned} {x}'_{2{D}} ={V}\ \mathrm{where}\ {Z}_{{D}} \equiv {Z}(\tau +\tau _1 +\tau _2 )={H}(\tau _2 ){Z}_{{C}}\nonumber \\ \end{aligned}$$

(42)

Let us assume that at this time, we have:

$$\begin{aligned} {x}'_{1{D}}<{V},\chi \left| {{x}_{2{D}} -{x}_{1{D}} } \right| <{u}_{{r}2} \end{aligned}$$

(43)

For $0<{t}-\tau -\tau _1 -\tau _2 <\tau _3$, the system undergoes a slip–stick motion (${x}'_1 <{V},{x}'_2={V}$). A periodic motion of period ${T}=\tau +\tau _1 +\tau _2 +\tau _3$ is obtained if we have:

$$\begin{aligned} {Z}_0 =\varGamma (\tau _3 ){Z}_{{D}} \end{aligned}$$

(44)

This motion depends on 5 parameters (${x}_{10} ,\tau ,\tau _1, \tau _2 ,\tau _3$). Assuming that ${u}_{{r}1}$ is defined by (41), these parameters are subjected to 5 scalar conditions related to (38) and (44), taking into account that (42) is included in (44).

Example: For

$$\begin{aligned} \eta= & {} 4.2,\chi =0.8,\,{V}=0.2186,{u}_{{s}1}\nonumber \\= & {} 0.1278,\,{u}_{{s}2} =0.0219,\,{u}_{{r}2} =0.4 \end{aligned}$$

(45)

we obtain:

$$\begin{aligned} \tau= & {} 0.4845,\,\tau _1 =0.2,\,\tau _2 =0.78,\,\tau _3 =2.809,\,{u}_{{r}1}\nonumber \\= & {} 0.1921,{z}_{10} =-\,0.2134,\,{z}_{20} =0.2243 \end{aligned}$$

(46)

The phase portraits of the system are shown in Figs. 9 and 10.

Fig. 10

Phase portrait of the colliding mass

Moreover, the constraints during the sticking motion of ${m}_1 (0<{t}-\tau <\tau _1 )$ and the constraints during the sticking motion of ${m}_2 (0<{t}-\tau -\tau _1 -\tau _2 <\tau _3 )$ are fulfilled.

7 Concluding remarks

In this work, a two-degree-of-freedom oscillator excited by a moving base with constant velocity and colliding with a rigid obstacle is considered. This system is strongly nonlinear; however, assuming that the dry friction forces are given by the Coulomb’s laws and assuming perfect elastic impact when a collision with the obstacle occurs, several sets of periodic motions including some phases of slip motion, overshooting motion, sticking motion and impact are found in analytic form.

8 Appendix: Symmetry of the solution

For ${T}/2<{t}<{T}$, the motion is defined by

$$\begin{aligned} {Z}({t})= & {} {H}({t}){H}_0 {Z}_0 ,{Z}_0 =\left( {\begin{array}{l} {z}_0 \\ {z}'_0 \\ \end{array}} \right) \nonumber \\ {z}'_0= & {} \left( {\begin{array}{l} 0 \\ {z}'_{20} \\ \end{array}} \right) ,{Z}({T})={H}({T}){H}_0 {Z}_0 ={Z}_0 \end{aligned}$$

(47)

From (47) it results

$$\begin{aligned} {Z}({T}-{t})={H}({T}-{t}){H}_0 {Z}_0 \end{aligned}$$

(48)

From the following properties [13] of the matrix H(t):

$$\begin{aligned} {H}({T}-{t})= & {} {H}(-{t}){H}({T}) \nonumber \\ {H}(-{t})= & {} {FH}({t}){F},{F}=\left( {\begin{array}{ll} {I} &{} \quad 0 \\ 0 &{} \quad -{I} \\ \end{array}} \right) \end{aligned}$$

(49)

we deduce

$$\begin{aligned} {Z}({T}-{t})={FH}({t}){FZ}_0 \end{aligned}$$

(50)

Taking into account that ${z}'_{10} =0$, ${FZ}_0 ={H}_0 {Z}_0$. It results:

$$\begin{aligned} {Z}({T}-{t})={FZ}({t}) \end{aligned}$$

(51)

The phase portrait of the first mass is symmetric with respect to the line

${z}'_1 =0$ and the phase portrait of the second mass is symmetric with the line ${z}'_2 =0$.